[ExI] [Extropolis] Gödel and physical reality
William Flynn Wallace
foozler83 at gmail.com
Wed Nov 3 18:15:05 UTC 2021
no finite (human invented) theory can ever answer every question or
provide every property of these mathematical objects. Who says so?
Nobody asked me! I can answer any question whatsoever, because of my
extended reading into everything. Oh, wait, maybe you meant *correct*
answers? bill w
On Wed, Nov 3, 2021 at 12:57 PM Jason Resch via extropy-chat <
extropy-chat at lists.extropy.org> wrote:
>
>
> On Wed, Nov 3, 2021 at 11:09 AM Giulio Prisco via extropy-chat <
> extropy-chat at lists.extropy.org> wrote:
>
>> On Wed, Nov 3, 2021 at 4:59 PM John Clark <johnkclark at gmail.com> wrote:
>> >
>> > On Wed, Nov 3, 2021 at 10:38 AM Giulio Prisco <giulio at gmail.com> wrote:
>> >
>> >> > Gödel's result implies an independence of mathematical truth from
>> any human theory of that truth, and I would argue, even makes mathematical
>> truth independent of the physical universe.
>> >
>> >
>> > Mathematics is the language of physics but like any language it can be
>> used to write both fiction and nonfiction. English is also a language and
>> it can be used to write a book about Quantum Mechanics but it can also be
>> used to write a book about Harry Potter, and both books could be equally
>> grammatically correct. I sometimes have the feeling that some of the more
>> abstruse areas of modern mathematics might be the equivalent of
>> mathematical Harry Potter novels.
>>
>
> We have the freedom to choose and make up mathematical theories, just as
> we have the freedom to choose and make up physical theories. But we don't
> bother with the ones that give nonsensical or wrong answers. We want our
> mathematical theories to be able to prove things that are objectively
> verifiable, such as whether or not a given program will halt or not. If a
> mathematical theory failed to give a correct prediction to such a question
> we would discard it, just as we would a physical theory that gave bad
> predictions. I think we ought to not confuse the theories (human invented)
> from the objects we try to describe using those theories. It seems you lump
> both the theories and the objects together when you say mathematics, but
> you don't make this same mistake when you speak of physical theories vs.
> physical objects. I think Gödel's incompleteness result effectively proves
> mathematical objects are necessarily distinct from any theory attempting to
> describe them, for no finite (human invented) theory can ever answer every
> question or provide every property of these mathematical objects.
>
>
>> >
>> >> > But it can also be argued that it makes mathematical truth *strongly
>> >> dependent* on the physical universe.
>> >
>> >
>> > That's why I would argue physics is more fundamental than mathematics,
>> and that's why a book on advanced computational theory sitting on a shelf
>> cannot add 2+2, the ideas in the book have to be implemented in physics
>> before it can actually do anything. If there is an even number that is not
>> the sum of two prime numbers but the number is so huge that it cannot be
>> calculated by physics even in theory then I think it would be meaningless
>> to say they're really "is" such a number.
>>
>
> Consider the existence of some little "game of life" universe that is
> 1,000 x 1,000 cells, and there is a simple automaton in there with a
> rudimentary nervous system. It can contemplate 2+2=4, but its brain and
> Game of Life universe is too small for it to conceive of numbers larger
> than a million. What effect, if any, would you say the multiverse theory
> has for the meaninglessness or meaningfulness of different numbers, which
> might be comprehensible in some universes but not in others? If the
> Goldbach counterexample is too large to represent in this universe, but not
> others, does it then make sense to say there is such a number? If not, what
> role does physical causality/interconnectedness have to do with existence
> or non-existence of numbers?
>
>
>> >
>> I agree, physics is more fundamental than mathematics.
>>
>> >> > our universe lacks the computational and memory resources to even
>> list all the properties of the number "3". In this sense, the number 3 is a
>> larger, and more complex object than our physical universe, for there are
>> an infinite number of true statements concerning the number 3,
>> >
>> >
>> > That depends on exactly how you define "complexity", but you wouldn't
>> need to list everything the number 3 can do to uniquely defined it, but if
>> you define the complexity of something as the minimum number of characters
>> needed to define it then the first billion digits of π are more complex
>> than all of the digits of π since all the digits of π are uniquely defined
>> by just 24 characters, 4*(1-1/3+1/5-1/7+1/9 -...), but you'd need 1
>> billion characters to uniquely define just the first billion characters of
>> π.
>>
>
> We can point to "3" quite easily, but to perfectly define 3 requires the
> ability to specify all of 3's properties. Many of 3's properties are
> unknowable to us with our finite mathematical theories. For example, is "3"
> more or less than the number of Goldbach counterexamples? We don't
> currently know. It may not even be knowable using our existing mathematics.
> In fact, it could require a mathematical theory with more than 10^120
> axioms to decide. There will always be such properties of 3 that are
> unknowable/unanswerable, no matter how big and complex a theory of
> mathematics we develop, so in a very real sense, "3" is bigger than our
> universe, and much bigger than the information complexity of the first
> billion characters of π. Actually "π" is a perfect example of how
> mathematical objects can have greater information content than the
> universe, you can't even write down the digits of π in our universe. Where
> then, do the digits of π exist?
>
>
>> >
>> >> > The finiteness of our universe also means there are axiomatic
>> systems which contain more axioms than there are atoms in our universe, so
>> we could never conceive of them.
>> >
>> >
>> > A logical system is only as strong as its weakest axiom, that's why
>> axioms need to be simple and intuitively obvious.
>
>
> Principles in physics don't need to be intuitively obvious. I think it may
> be a mistake to require the same for mathematical axioms. It may be that
> very counterintuitive axioms are required to build more powerful and
> effective mathematics, which should be the only and true test of a
> mathematical theory.
>
>
>
>> We could just add the Goldbach Conjecture as an axiom but it's truth is
>> not intuitively obvious and it would be very embarrassing if after doing
>> this a computer crunching through huge numbers happen to find an even
>> number that was not the sum of two prime numbers, it would mean all the
>> work done in mathematics since the axiom had been added was meaningless.
>>
>
> Yes, mathematicians tend to be conservative when it comes to adding or
> changing axioms, ann not altogether without good reason, but I think Gödel
> was on to something when he said empiricism has a place in mathematics.
>
> Jason
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